Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,022 questions
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Karhunen–Loève approximation of Brownian motion and diffusions
The Karhunen–Loève theorem says that Brownian motion on the interval [0,1] can be represented as follows:
$W_t = \sum_{n=1}^\infty Z_n \frac{\sin((n-1/2)\pi t)}{(n-1/2)\pi},$
where $Z_n \sim \mathcal{...
- 1,666
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3
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293
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How can we pave the multiplicative semigroup $(\mathbb N,\cdot)$?
Let $(S,\cdot)$ be a semigroup and $W\subseteq S$ be a subset. Let me call $W$ "tile" if the following property is satisfied: there exist $s_1,...s_k\in S$ such that the sets $s_i\cdot W$ are pairwise ...
- 6,034
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1
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A Question on Random Matrices
Consider the following $n\times n$ random matrix $V_{n}$ where the $(p,q)$ entry is given by
$$
V_{n}(p,q):= \frac{1}{\sqrt{n}}\exp(2\pi i(p-1) x_{q})
$$
where $x_{1},x_{2},\ldots,x_{n}$ are iid ...
- 3,626
4
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2
answers
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Examples of Lyapunov functions for Markov processes
I am reading about Lyapunov functions for Markov processes, and I am having trouble thinking of examples to keep in mind as I read. If $X_t$ is a continuous-time Markov process with generator $L$, a ...
- 29.8k
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4
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3k
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What is the cover time of a random walk on a cube?
I can't quite figure this problem yet. There is an ant at one vertex of a cube. The ant goes from one vertex to another by choosing one of the neighboring vertices uniformly at random. What is the ...
- 241
11
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1
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588
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Wiener Sausages in Riemann Surfaces
Let $M$ be a Riemann surface (or a higher dimensional manifold) and let's assume that it's geodesically complete. Let $W(t)$ be a Brownian motion on the surface accordingly to the manifold's Laplacian ...
- 3,626
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votes
2
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200
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Good probability measues on $S^1$ reprented by a kernel
I was looking for some good references for properties/theorems/characterizations of 'good/important' probability measures on the unit circle $S^1$ ( and/or on spheres $S^n$ ).In particular, I want ...
- 1,471
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2
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Assigning positive edge weights to a graph so that the weight incident to each vertex is 1.
Let $\Gamma=(G,E)$ be a connected undirected graph, with no loops or multiple edges. $G$ is finite or countably infinite. For each edge $e=\{x,y\}\in E$, we assign a positive, symmetric edge weight $...
- 269
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2
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Weierstrass' function and Brownian motion
Is there a known connection between Weierstrass' function
$W_\alpha (x) = \sum_{n=0}^\infty b^{- n \alpha} \cos(b^n x)$
and Brownian motion? Specifically, when $\alpha = 1/2$, the Weierstrass ...
- 1,666
0
votes
1
answer
292
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Probability of preserving connectivity between pair of vertices in weighted graph
Let $G=(V,E)$ be an undirected graph and $p \colon E \mapsto (0,1]$ defines weights of its edges.
Let's fix two connected vertices $v_1, v_2 \in V$.
Random graph $G'=(V,E')$ is obtained from $G$ by ...
- 3
1
vote
1
answer
868
views
optimal coupling
Given probability distributions $(\mu_1, \ldots, \mu_n)$ on a nice state space $E$ is it always possible to find a random vector $(X_1, \ldots, X_n)$ such that $(X_k, X_{k+1})$ is an optimal coupling ...
- 2,133
3
votes
1
answer
412
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Sparse representation of a distribution with independent and correlated variables
Here's what I'm trying to do:
Imagine a probability distribution over $\mathbf{R}^2$, $P(x,y)$. I can approximate $P(x,y)$ with set of $N$ points $\{(x,y)_i\}$ drawn from $P$. By approximate, I mean ...
- 1,902
2
votes
2
answers
861
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Spectral gap of a product of Markov processes
For $m \in [N] \equiv \{1,\dots, N\}$, let $Q^{(m)}$ be the generator of a (well-behaved) continuous-time Markov process on a finite state space $[n_m]$. Write $J \equiv (j_1,\dots,j_N) \in \prod_m [...
- 15.4k
5
votes
2
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679
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distance in terms of the variance between two absolutely continuous probability measures
Consider two probability measures $\mu_0$ and $\mu_1$ on $\mathbb{R}^n$, such that $\mu_0\ll \mu_1$. Then I can define a "distance" like quantitiy
$$
\mathrm{Var}_{\mu_1}\left(\frac{\mathrm{d}\mu_0}{\...
- 1,081
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3
answers
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Markov random field with continuous index set
Hi
There's Markov random field (MRF) which, by my Wikipedia-based knowledge, is an extension of Markov chain. I'd like to think of it as going from 1D to higher dimensional spaces. Inherent in its ...
- 355
2
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0
answers
281
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Is the variance of an eigenfunction of a finite state space aperiodic irreducible markov chain starting at a single state always non-decreasing?
I am reposting a previous question due to incorrect initial formulation.
Given an ergodic (aperiodic and irreducible) finite state space Markov chain $P$. Let $f$ be an eigenfunction, i.e., $P_t f = ...
- 4,466
1
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0
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172
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Relative convergence of probability measures
I want to define a fucntional on the space of random variables continuous with respect to small relative perturbations of the underlying probability measure. To explain this, consider probability ...
- 11
3
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1
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574
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is the variance of a test function of a markov chain always increasing?
Edits: Changed function to eigenfunction. I should have stated the problem with more explicit conditions. Anyways I realized the original formulation is not true, even when one starts at a single ...
- 4,466
4
votes
2
answers
350
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analogue of GUE and Ginibre in higher dimensions
This is a completely unmotivated question, but what happens to the 1-point marginal distribution for the following $N$-point joint distribution:
$$\displaystyle p(z_1,\ldots, z_N) = C_N \exp\left(-\...
- 4,466
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0
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295
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Is there an idempotent measure on the free LD system?
This is a follow up question to MO question "Idempotent measures on the free binary system?".
Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law:
...
- 3,547
2
votes
1
answer
355
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Why doesn't the argument of circular law convergence of Ginibre spectrum give the same result for GUE?
It appears I am profoundly confused in the following nice argument of Ginibre and Mehta and beautifully presented in Djalil Chafai's blog http://blog.djalil.chafai.net/2010/11/02/aspects-of-the-...
- 4,466
13
votes
1
answer
736
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Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 3,547
17
votes
4
answers
6k
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Correlated Brownian motion and Poisson process
Is there an (easy) way to construct, on the same filtered probability space,a Brownian motion $W$ and a Poisson process $N$, such that $W$ and $N$ are not independent ?
I first asked this question ...
- 368
4
votes
1
answer
496
views
Correlation structure among the maximums of a Brownian motion
Is there a known correlation structure among the maximums of a Brownian motion on disjoint intervals ?
Let $(W_t)_{t\geq 0}$ be a one-dimensional standard Brownian motion,
and take the partition $0=...
- 328
9
votes
2
answers
1k
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Anti-concentration about the mean for sum of Bernoulli random variables
Let $w\ll n$ (say $w=n^{0.1}$) and $a_1,\ldots,a_w$ be positive real numbers such that $\sum_{i \in w} a_i=n$. Also, let $x_1,x_2,\ldots, x_w$ be i.i.d. $\pm 1$ random variables. What is the best $t$ ...
- 563
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2
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673
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"Probabilistic ultrafilters?"
A naive question.
Let $S$ be a set and let $[0,1]^S$ the set of functions from $S$ to the closed interval $[0,1]$.
Suppose given some function $P \colon [0,1]^S \to [0,1]$ satisfying the following ...
- 19.2k
10
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0
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809
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Where can I find analogues of combinatorial central limit theorems for other groups
The statement of Hoeffding's combinatorial central limit theorem is as follows: given for each $n$, an $n \times n$ matrix $A = (a_{ij})$, one can consider the random diagonal sum:
$$\displaystyle f(\...
- 4,466
6
votes
0
answers
189
views
average Riemannian distance between Identiity and a random point in SO(n) or SU(n)
I can compute the even moments of the Riemannian distance $d(Id, U)$ between the identity element and a uniformly chosen point on say $SU(n)$. But the odd moments elude me. Basically one needs to ...
- 4,466
1
vote
2
answers
316
views
Martingale part of the discontinuous put payoff
I need the martingale part of the put payoff (not $C^2$..). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$
$d[(S_t -K)^+ ]$ ??
I guess I need to use local times but how?
- 111
5
votes
1
answer
498
views
Percolation in Cayley graphs of semigroups.
Percolation in Cayley graphs of groups are studied by many researchers. There are also the concept Cayley graphs for semigroups. Are there any research about percolation in Cayley graphs for ...
- 6,201
10
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0
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391
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Question from an economist: solving a model of traders' behavior with expectations about the future values of the variable they are currently optimizing
Motivation
I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on ...
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5
votes
1
answer
2k
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Random points in a rectangular grid defining a closed path
Suppose we have a $n\times m$ rectangular grid (namely: $nm$ points disposed as a matrix with $n$ rows and $m$ columns).
We randomly pick $h$ different points in the grid, where every point is ...
- 51
8
votes
1
answer
1k
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Filtrations generated by cadlag martingales.
Let $(\Omega,P,\mathcal{F})$ be a probability space with filtration $\mathbb{F} = (\mathcal{F}_t), t \in [0,T]$, where $T$ can be finite or infinite. Let $M$ be a cadlag (local) martingale with ...
- 943
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2
answers
361
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Hausdorff dimension of non-recurrent walks
Preface: I am fairly new to the concept of Hausdorff dimension, so I don't know how interesting a question this is.
Identify walks on $\mathbb{Z}$ with infinite binary sequences (say $0$ means moving ...
- 53
11
votes
1
answer
835
views
Generalized Euclidean TSP
Suppose I have n sets $X_1,\dots,X_n$ consisting of $k$ points each, where all $nk$ points are i.i.d. uniform random samples in the unit square $[0,1]\times[0,1]$. Consider the shortest path that ...
- 1,125
8
votes
1
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969
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Probabilities independent of ZFC?
Hi guys,
is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of ZFC?
...
- 165
1
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0
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301
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Inverse Skorokhod Embedding Problem
The Skorokhod Embedding Problem is well known and has many documented solutions in the literature.
Now if we are given a Brownian stochastic basis (satisfying usual hypothesis), a diffusion $X_t$ (...
- 1,334
5
votes
1
answer
394
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Product of coordinates of a random point from Hamming sphere
Let us consider a boolean hypercube $C = \{-1, 1\}^n$. Let $S = \{x \in C \mid |\{i \mid x_i = -1\}| = \varepsilon n\}$ be a Hamming sphere in $C$ (here $\varepsilon$ stands for the fixed parameter ...
- 1,791
19
votes
0
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682
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support of the coupling between two probability measures
Given two Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}$, let $\Pi(\mu, \nu)$ denote all couplings between them, i.e., all Borel probability measures on $\mathbb{R}^2$ such that the ...
- 1,839
2
votes
1
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889
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Probability Measures and Cardinality > c
Is it possible to place non-trivial probability measures on sets of cardinality strictly greater than the continuum -- in particular, on sets of cardinality 2^c? (Any references would be appreciated.)...
- 33
9
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1
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695
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Probability of return vs. probability of return in minimal number of steps
Consider a random walk on a connected graph $G=(V,E)$. That is, associate to each neighbouring nodes $a,b\in V\ $ transition probabilities $\mathbb{P}(a\rightarrow b), \mathbb{P}(b\rightarrow a) $ ...
- 965
3
votes
1
answer
1k
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Cumulative distribution function of hypergeometric distribution
Does anyone know a closed form or a good approximation of the cumulative distribution function of hypergeometric distribution?
- 177
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vote
2
answers
498
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infimum of a set of positive r.v. with the same distribution
Let $Y$ be real valued random variable on probability space $(\Omega,
\mathcal{F}, P)$, such that $Y>0$ almost surely. Suppose $(X^a: a\in
\Lambda)$ be a set of random
variables in the same ...
- 1,399
1
vote
1
answer
1k
views
infimum of a set of stopping times
Let $(Y^a: a\in \Lambda)$ be a set of random processes given by
$$Y^a(s) = \int_0^s \sigma^a(r) dW(r)$$
where $W$ is Brownian motion w.r.t. filtered probability space
$(\Omega, \mathcal{F}, P, \...
- 1,399
24
votes
1
answer
2k
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A puzzle about finding three points $(x,y)$, $(x,z)$ and $(y,z)$ in a subset of a square.
I was asked (by myself) to give a proof of the following seemingly simple geometric statement, but after thinking a little I now suspect it could be less elementary than I thought (or am I being silly?...
- 60.5k
6
votes
2
answers
729
views
Has the following kind of (minimum degree $d$) random graph been studied?
The following random construction is simple enough that I am guessing it must have been studied. Fix $d \ge 3$, and let $n > d$. For each of the $n$ vertices, pick exactly $d$ other vertices to ...
- 7,857
2
votes
1
answer
194
views
Is there any result discribing the value of the correlation of a measurable function of `$X$` and itself: `$corr(f(X),X)$` ?
Let $X$ be a random variable, and $f$ a measurable function. Is there any particular relationship between the expression of $f$ and $corr(f(X),X)$?
BACKGROUND
The background of asking the value of $...
- 23
4
votes
1
answer
3k
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The only continuous martingales with stationary increments are Brownian motions
I know that the above statement is true, but I can't demonstrate it.
It's a pretty powerful theorem, here is its mathematical formulation:
Theorem: The only continuous martingales with stationary ...
- 41
-2
votes
1
answer
248
views
for examples in probability [closed]
Give an example satisfying the following conditions:
give out a sequence of random variables defined on a probability space, and a sub sigma algebra: the sequence converges almost surely to a limit ...
- 105
3
votes
2
answers
994
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measurability of integrated functions
DISCLAIMER: I'm not a mathematician, but a computer scientist, so I hope the question is not trivial (or perhaps I hope so, in order to get a definitive answer). Anyway it's not a homework, as ...
user11618